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Solve each equation by factoring. Check your answers.
1. x 2 + 6x + 8 = 0 –4, –2 2. x 2 + 18 = 9x 3, 6
3. 2x 2 - x = 3 –1, 32
4. x 2 - 10x + 25 = 0 5
5. 2x 2 + 6x = -4 –2, –1 6. 3x 2 = 16x + 12
–23 , 6
Solve each equation by finding square roots.
7. 5x 2 = 80 –4, 4
8. x 2 - 4 = 0 –2, 2
9. 2x 2 = 32 –4, 4
10. 9x 2 = 25 –53 , 53
11. 3x 2 - 15 = 0
12. 5x 2 - 40 = 0
–"5 , "5
–2 "2, 2 "2
Solve each equation by factoring or by taking square roots.
13. x 2 - 4x = 0 0, 4
14. 6x 2 + 4x = 0 –23 , 0
16. 3x 2 = 48 –4, 4
17. 2x 2 = 8x –0, 4,
15. 12 x 2 - 147 = 0
–72 , 72
18. 4x 2 - 80 = 0
–2 "5, 2 "5
19. Firefighters A smoke jumper jumps from a plane that is 1700 ft above the
ground. The function y = -16t 2 + 1700 gives the jumper’s height y in feet
at t seconds.
about 6.61 s
a. How long is the jumper in free fall if the parachute opens at 1000 ft?
b. How long is the jumper in free fall if the parachute opens at 940 ft?
about 6.89 s
Solve each equation using tables. Give each answer to at most two decimal places.
20–31. See margin.
20. x 2 + 5x + 3 = 0
21. x 2 - 7x = 11
22. 2x 2 - x = 2
golden rectangle?
62. Matrices Find the possible values of x and y. (A matrix with exponent 2 means
33. Multiple Choice The
that you multiply the matrix by itself.)
period of a pendulum is
2 the
xthe2time
22pendulum
10
x ≠ 4, y ≠ 1 or x ≠ –4, y ≠ 9
c takesd to=swing
c
d
2
3 y
15 back
j and forth. The function O = 0.81t relates the length O in
feet of a pendulum to the time t in seconds that it takes to swing back and forth.
63. Using
tables, how center
might you
recognize
that a has
quadratic
equation
likelyin
has
The convention
in Portland,
Oregon,
the longest
pendulum
the
exactly
one
solution?
no solutions?
United
States.
The pendulum’s
length is 90 ft. Find the period. B
8.5 seconds
10.5
90inseconds
=seconds
0 can be written
factored form111
as seconds
64. The equation
x 2 - 10x + 24
o
E
E
u
l
T
t
t
(xOpen-Ended
- 4)(x - 6) Write
= 0. How
can youinuse
this factform
to find
of by
34.
an equation
standard
thatthe
youvertex
can solve
- 10x +that
24?you
thefactoring
graph ofand
y =an
x 2equation
See
backsolve
of book.
cannot
by factoring.
Check students’ work.
65. Physics Suppose you throw a ball straight up from the ground with a velocity
of 80 ft/s. As the ball moves upward, gravity slows it. Eventually the ball begins
atic Equations
Functions
to falland
back
to the ground. The height h of the ball after t seconds in the air is
given by the quadratic function h(t) = -16t 2 + 80t.
a. How
high does
the ball go? 100 ft
25. –0.94,
2.34
30. –5.16, 1.16
b. For how many seconds is the ball in the air before it hits the ground? 5 s
26. –5.53, 0.36
31. –1.16, 2.16
66. a. Let a . 0. Use algebraic or arithmetic ideas to explain why the lowest
27. –1,
0.25
point
on the
graph of y = a(x - h)2 + k must occur when x = h.
b. Suppose
that 5.12
the function in part (a) is y = a(x - h)3 + k. Is your
28. –3.12,
reasoning still valid? Explain. a–b. See back of book.
5
5
5
29.–1.46, 5.46
, Web Code: aga-0505
Lesson 5-5 Quadratic Equations
271
5
Test Prep
Multiple Choice
67. What are the values of x that satisfy the equation 3 - 27x2 = 0? B
1
A. x = 43
B. x = 43
1
C. x = 1
D. x = 2!6 or x = 22!6
9 or x = 2 9
68. What are the solutions of the equation 6x2 + 9x - 15 = 0? G
F. 1, -15
G. 1, 2 5
2
5
J. 3, 2
69. For which equation is -3 NOT a solution? D
B. x2 - 21 = 4x
A. x2 - 2x - 15 = 0
C. 2x2 + 12x = -18
D. 9 + x2 = 0
H. -1, -5
76. [2] (x – 3)(x ± 4) ≠ 0,
or x2 ± x – 12 ≠ 0,
or any constant
multiple of this
equation
[1] the correct
factoring, not written
as an equation
77. [2] (x – a)(x – 5) ≠
x2 ± bx – 20
5a ≠ –20
a ≠ –4
The other solution is
–4.
[1] –4, with no work
shown
Short Response
70. What are the solutions of the equation (2x - 7)2 = 25? G
F. 6, -6
G. 6, 1
H. 6, -1
J. -6, -1
71. Find the sum of the solutions to the equation x2 + 2x - 15 = 0. D
A. 8
B. -8
C. 2
D. -2
72. Find the product of the solutions to the equation x2 - 8x = 9. J
F. 6
G. -6
H. 9
J. -9
73. Which equation has 2 2
5 as a solution? D
A. (2x - 5)(x + 1) = 0
C. (5x - 2)(x + 1) = 0
B. (2x + 5)(x + 1) = 0
D. (5x + 2)(x + 1) = 0
74. How many times does the graph of y= x2 - 4x + 5 cross the x-axis? F
F. 0
G. 1
H. 2
J. 33
75. The equation x2 - 3x + a = 0 has two roots. One root of the equation is 2.
What is the other root? C
A. -2
B. -1
C. 1
D. 3
76. A quadratic equation has solutions 3 and -4. Write a possible equation.
76–77. See left.
77. One solution to the equation x2 + bx - 20 = 0 is 5. Find the other
solution.